Descartes' Rule of Signs Revisited

As it has come to be stated, if a real polynomial is arranged in ascending or descending powers, its number of positive roots is no more than the number of sign variations in consecutive coefficients, and differs from this upper bound by an even integer. In 1807, Budan extended Descartes' Rule of Signs to determine an upper bound on the number of real roots in any given interval (p, q). It extends Descartes' Rule of Signs by substituting x' = x p and x" = x q and counting the sign variations lost in the sequence of coefficients between the resulting transformed polynomials. This forms the upper bound; the actual number of roots differs by an even number. Budan's theorem, largely ignored, was restated by Fourier in 1820 and this better known formulation is usually referred to as the Fourier-Budan Theorem. In 1829 Descartes' Rule of Signs and related estimates were seemingly eclipsed by Sturm's Sign Sequence Theorem, which provides a precise measure for how many real roots lie in any interval. Nevertheless, due to the computational complexity of generating the Sturm sequences, less precise estimates like Descartes' remain useful to this day. Budan's theorem is used, for example, in an efficient computer algebra algorithm called the Continued Fractions Method for Isolation of the Real Roots of an Equation [2]. Because Descartes' Rule of Signs is such a useful and classic result, it is surprising that the available literature does not seem to indicate whether its upper bound is sharp: Given a sign sequence, can one always find a polynomial with the maximum possible number of positive roots and whose coefficients have the given sign sequence? Further, can one always find polynomials with the given sign sequence and the other allowable numbers of positive roots, namely those that are any even number less than the upper bound? We limit our consideration to polynomials with all coefficients non-zero, leaving the case of vanishing coefficients as an open problem. We show that the upper bound can always be attained for any sign sequence of coefficients that contains no zeros, and provide an explicit formula for finding such a polynomial. We then show that this polynomial can be modified to reduce the number of roots by any even number while maintaining the same sign sequence. Thus Descartes' Rule of Signs is "complete," at least for sign sequences with no zeros.